Heat-flux positivity and monotonicity of TGLF in the gradients
The brief asks for physics structure in the transport surrogates: zero heat flux below the instability threshold, no negative heat flux, and a heat flux that does not fall when its own temperature gradient rises. Without that structure a surrogate can return fluxes the transport solver treats as unphysical. The owner decided in OQ-19 to put a ReLU on the electron and ion heat fluxes, \(Q_e\) and \(Q_i\), and to measure monotonicity on TGLF before imposing it.
Positivity: a clip in use, a penalty in training
A ReLU inside the network passes no gradient for points it predicts as stable, so those points stop training. We follow QLKNN’s approach instead: the output stays linear during training, and where TGLF’s flux is zero or negative the loss only penalises a prediction above a small negative margin. In use, each member’s \(Q_e\) and \(Q_i\) are clipped at zero. The particle fluxes keep their sign; inward particle flux is physical.
The cost is that TGLF’s own negative heat fluxes cannot be represented: Q <= 0 in 4.6% (MAST-U) and 5.2% (ITER) of cases for Q_e, 11.8% and 13.2% for Q_i; Q < -1 gB in 0.8% and 0.7% (Q_e) and 6.5% and 4.3% (Q_i) of cases with |Q| > 1 gB; Q_e + Q_i changes by more than 10% in 0.9% and 1.0% of casesC-104. This is ceiling K-021. TORAX converts the flux to a diffusivity and clips that at a minimum, so the coupled loop already discards negative heat fluxes.
To choose the training rule we ran short CPU trainings of the TGLF-NN network on a fifth of the v1 training data, one per rule and machine. The measure is how often the surrogate returns exactly zero where TGLF’s flux is zero or negative, on the v1 test split:
| training of \(Q_e\), \(Q_i\) | exact zero where TGLF’s \(Q \le 0\) |
|---|---|
| linear outputs, clipped afterwards | Q_e 0.13 (MAST-U) and 0.17 (ITER); Q_i 0.16 and 0.20C-105 |
| fitted to \(\max(Q, 0)\) | Q_e 0.10 (MAST-U) and 0.15 (ITER); Q_i 0.06 and 0.06C-106 |
| stability penalty, margin 0 | Q_e 0.18 (MAST-U) and 0.21 (ITER); Q_i 0.13 and 0.17C-107 |
| stability penalty, margin 1 (adopted) | Q_e 0.28 (MAST-U) and 0.40 (ITER); Q_i 0.25 and 0.43C-108 |
| stability penalty, margin 0.5, weight 10 | Q_e 0.45 (MAST-U) and 0.62 (ITER); Q_i 0.48 and 0.63C-109 |
Fitting \(\max(Q, 0)\) does worse than clipping afterwards. A larger penalty gives more zeros but starts to zero real fluxes and costs accuracy (Q_i median relative error +0.03 (MAST-U) and +0.05 (ITER); 1.0% and 1.3% of cases with Q_i > 1 gB predicted as zeroC-111). The adopted setting costs little: Q_i median relative error +0.005 (MAST-U) and +0.009 (ITER), Q_e within ±0.004; stable/unstable agreement +0.009 and +0.016; Q > 1 gB predicted zero in 0.1% (Q_e) and 0.3% (Q_i) of such casesC-110. These are smoke runs, smaller than the production models, and are used to rank the rules, not to quote accuracy. The production variant (tglfnn_symlog_relu) is to be trained on GPUs and compared with the M3 results of entry 18.
Monotonicity: measured on TGLF first
We scanned TGLF directly. Per machine, 35 points from the v1 test split with both heat fluxes above 1 gB were each scanned over \(a/L_{T_e}\) and over \(a/L_{T_i}\), all other inputs fixed: 560 cases per machine, all completed; the 70 re-run base cases per machine reproduce the dataset’s fluxes bit for bitC-112. TGLF is deterministic here, so any fall in the flux along a scan is TGLF’s behaviour, not noise. The flux falls below its running maximum along most scans: falls somewhere in 77% (Q_e) and 80% (Q_i) of MAST-U scans, 51% and 74% of ITER scans; by more than half in 40% and 57% (MAST-U), 34% and 57% (ITER)C-113. The v1 ensemble, evaluated on the same scans, falls less often (falls somewhere in 20% (Q_e) and 49% (Q_i) of MAST-U scans, 23% and 37% of ITER scans; by more than half in 3% and 23% (MAST-U), 9% and 17% (ITER)C-114), and its slope is negative at a minority of test points (Q_e 8% (MAST-U) and 11% (ITER); Q_i 25% and 22%C-115). The ensemble is smoother than TGLF, so its non-monotone regions are not the same as TGLF’s.
We have not established why TGLF falls along these scans: whether the drops are changes of the dominant mode, or come from scanning one input with the others fixed, is not measured.
Decision OQ-20
The owner was asked whether to drop monotonicity as a requirement, since TGLF does not satisfy it on these points. The decision (OQ-20, 1): monotonicity stays a requirement for the TGLF surrogates, \(Q_e\) non-decreasing in \(a/L_{T_e}\) and \(Q_i\) non-decreasing in \(a/L_{T_i}\). Where TGLF decreases, the surrogate departs from TGLF by construction, and that departure is to be measured and reported as a ceiling. The same decision widened the ITER collisionality range for dataset v3 (OQ-20, 2).
M3 is in progress: the production ReLU variants, the monotone surrogate and the measured departure from TGLF are next.
Technical details → Surrogates: heat-flux sign, monotonicity. Decisions → OQ-19, OQ-20.
Technical details → Surrogates › heat flux sign · Surrogates › monotonicity
Decisions → OQ-19 · OQ-20